Vortex pairs on surfaces (original) (raw)
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Vortex pairs on a triaxial ellipsoid and Kimura's conjecture
Journal of Geometric Mechanics, 2018
We consider the problem of point vortices moving on the surface of a triaxial ellipsoid. Following Hally's approach, we obtain the equations of motion by constructing a conformal map from the ellipsoid into the sphere and composing with stereographic projection. We focus on the case of a pair of opposite vortices. Our approach is validated by testing a prediction by Kimura that a (infinitesimally close) vortex dipole follows the geodesic flow. Poincaré sections suggest that the global flow is non-integrable.
Vortices on surfaces - final version
It was recognized, since the seminal papers of Arnold and Ebin-Marsden , that Euler's equations are the right reduction of the geodesic flow in the group of volume preserving diffeomorphisms. In 1983 Marsden and Weinstein ([135]) went one step further, pointing out that vorticity evolves on a coadjoint orbit on the dual of the infinite dimensional Lie algebra consisting of divergence free vectorfields. Here we pursue a suggestion of that paper, namely, to present an intrinsic Hamiltonian formulation for a special coadjoit orbit, which contains the motion of N point vortices on a closed two dimensional surface S with Riemannian metric g. Our main results reformulate the problem on the plane, mainly C.C.Lin' s works ([126], , 1941) about vortex motion on multiply connected planar domains. Our main tool is the Green function G g (s, s o ) for the Laplace-Beltrami operator of (S, g), interpreted as the stream function produced by a unit point vortex at s o 2 S. Since the surface has no boundary, the vorticity distribution w has to satisfy the global condition RR S w W = 0, where W is the area form. Thus the Green function equation has to include a background of uniform counter-vorticity. As a consequence, vortex dynamics is affected by global geometry. Our formulation satisfies Kimura's requirement ([103], 1999) that a vortex dipole describes geodesic motion. A single vortex drifts on the surface, whose Hamiltonian is given by Robin's function, which in the the case of topological spheres is related to the Gaussian curvature . Results on numerical simulations on flat tori, the catenoid and in the triaxial ellipsoid are depicted. We present a number of questions, intending to connect point vortex streams on surfaces with questions from the mathematical mainstream.
2008
We consider NNN point vortices sjs_jsj of strengths kappaj\\kappa_jkappaj moving on a closed (compact, boundaryless, orientable) surface SSS with riemannian metric ggg. As far as we know, only the sphere or surfaces of revolution, the latter qualitatively, have been treated in the available literature. The aim of this note is to present an intrinsic geometric formulation for the general case.
The motion of point vortices on closed surfaces
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015
We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of...
Fluid Dynamics Research, 2003
Two-dimensional hydrodynamics is formulated geometrically with a metric deÿned in terms of stream function and vorticity for ows of an inviscid incompressible uid in unbounded space. Based on this, the vorticity equation is derived as a geodesic equation over a group of di eomorphisms of a uid at rest at inÿnity with vorticity having compact supports, which is also rewritten in the form of a Hamilton's equation with a Lie-Poisson bracket. In particular, this formulation is applied to a ow induced by a ÿnite number (N ) of point vortices. It is shown that the geodesic equation reduces to the well-known Hamiltonian system of N point vortices. This is considered as a dynamics of an internal space of ÿnite degrees of freedom, as against the former background dynamical system described by a group of di eomorphisms of inÿnite dimensions and Riemannian geometry.
The interaction of an elliptical patch with a point vortex
Fluid Dynamics Research, 2000
An integrable model is proposed to analyze the two-dimensional asymmetrical interaction of two vortices, either co-rotating or counter-rotating, in the absence of viscosity. To this purpose two assumptions are made: one vortex is uniform and elliptical and the other one is a point vortex. It follows a system with three degrees of freedom, for which ÿrst two integrals of the motion are known: the excess energy and the second-order moment of the vorticity ÿeld. By considering the latter as a parameter, the two remaining degrees of freedom are combined into a complex variable z, hence the isolines of the excess energy may be analyzed in the z-plane, to study the motion of the system. In particular, the number of the extremal points of the excess energy ÿeld, which identify the stationary conÿgurations of the system, is calculated in di erent regions of the parameter space. The excess energy ÿeld, associated to each of these regions, leads to the speciÿcation of the system dynamics for any possible initial condition. Depending on the values of the parameters and on the initial conditions, we ÿnd di erent types of motion, corresponding to periodic, merging (also for counter-rotating vortices) and diverging solutions. Diverging interactions lead to a kind of straining out of the patch and they are possible only for counter-rotating vortices, with the ratio between the circulation of the point vortex and the one of the patch equal to − 1 2. Particular attention is given to the interactions leading to merging, where the analysis in terms of an elliptical patch under rotating strain provides an useful physical interpretation.
Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability
Regular and Chaotic Dynamics, 2019
We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid E(a, b, c) : x 2 /a + y 2 /b + z 2 /c = 1, a < b < c. The equations of motion are transported to S 2 × S 2 via a conformal map that combines confocal quadric coordinates for the ellipsoid and sphero-conical coordinates in the sphere. The antipodal pairs form an invariant submanifold for the dynamics. We characterize the linear stability of the equilibrium pairs at the three axis endpoints.
On the interaction between a strong vortex pair and a free surface
Physics of fluids, 1990
The initial motion of a pair of strong vortices suddenly created close to a free surface is calculated analytically by means of a Taylor expansion in time. While weak vortices moving toward a free surface repel each other [see Hydrodynamics (Cambridge V.P., Cambridge, 1932)], strong vortices attract each other to the leading order. But, they cannot merge and annihilate each other because of nonlinear free-surface effects. Several features of the solution depend on whether or not the vortices are below a critical depth. At the critical depth the two vortex points span an angle of 120 0 with respect to the surface center. The dominating gravitational effects are investigated.